buck4155967-san,
that's difficult to explain
教訓 would be something you learn from a story or actual experience.
For example, your parents always tell you to prepare well for your exams, but you
don't take their words seriously as you believe you can do well anyway. You decide to
ignore your parents' advice and study only right before the exam to cover important
topics. You take the exam and find out you missed the actual "important topics" of exam
which you could easily realise that if you covered your study thoroughly.
The 教訓 from this experience would be "to listen to your parents" or could also be
"to study enough before the exam, not only betting where (which part of textbook) you should study".
Now 教え covers a bit wider meanings. This word literally means "teaching".
It can be, accordingly, any kind of lessons or teachings of something.
We often use this word 教え as "a way of living" and/or "spiritual teaching" as well.
For example, the way Japanese people appreciate any little things originate in 神道の教え
(i.e. teaching from Shintoism).
Hope this helps a bit
Natsuko(奈津子),
Team JapanesePod101.com